Overview of the LinearAlgebra Package

Transcription

1 verview of the LinearAlgebra Package Basic Functionality Description The LinearAlgebra package offers routines to construct and manipulate Matrices and Vectors, compute standard operations, query results and solve linear algebra problems. For a complete list of the routines in the LinearAlgebra package, see the Details of the Linear Algebra Package help page. utput Matrix, Vector, or an expression sequence of the two. Matrices 0 x 0 or smaller and vectors 0 x and smaller display the corresponding Matrix or Vector in the Maple worksheet. Matrices and vectors larger then this display a placeholder as output. To see the entries or structured views of the Matrix or Vector, double-click the placeholder. For more details, see the browse Matrix help page. Interfaces to the LinearAlgebra Package Commands Each command in the LinearAlgebra package is accessed by using either the long form or the short form of the command name in the command calling sequence. For more information, see the Using Packages help page. Long form LinearAlgebra[RandomMatrix](2); Short form with(linearalgebra): RandomMatrix(2); Maplets Some routines in the LinearAlgebra package come with Maplet interfaces. To see the available interfaces, see the Maplets[Examples][LinearAlgebra] help page.

2 Task Some routines in the LinearAlgebra package come with a task template to step you through the process of solving a linear algebra problem. For more information, see the Using Tasks help page. Student[LinearAlgebra] Package For students learning the concepts presented in an introductory linear algebra course, see the Student[LinearAlgebra] help page.

3 Essential LinearAlgebra Package Commands Basis return a basis for a vector space CharacteristicPolynomial construct the characteristic polynomial of a Matrix CrossProduct compute the cross product of two Vectors DeleteRow delete rows of a Matrix Determinant compute the determinant of a Matrix Dimension determine the dimension of a Matrix or a Vector DotProduct compute the dot product of two Vectors Eigenvalues compute the eigenvalues of a Matrix Eigenvectors compute the Eigenvectors of a Matrix GaussianElimination perform Gaussian elimination on a Matrix LeastSquares compute the least-squares to equations LinearSolve solve the linear equations A. x = b Map map a procedure onto an expression MatrixInverse compute the inverse of a square Matrix MatrixScalarMultiply compute the product of a Matrix and a scalar NullSpace compute a basis for the nullspace of a Matrix RandomMatrix construct a random Matrix ReducedRowEchelonForm perform Gauss-Jordan elimination on a Matrix SubMatrix construct a submatrix of a Matrix Transpose compute the transpose of a Matrix Examples with(linearalgebra): Construct a 5 x 5 Matrix. M:=RandomMatrix(5); M := K8 K98 K76 K4 29 K8 K77 K K8 57 K K2 69 K K9 K (4.) Construct a submatrix of the Matrix M, where the first list in the calling sequence selects corresponding row entries and the second list selects column entries. SubMatrix(M, [2..5], [2.., ]); K77 K72 K8 57 K2 K8 27 K2 87 K9 K74 (4.2) Construct the Sylvester Matrix of two polynomials. SylvesterMatrix(x^2+*x, 2*x); (4.) Compute the Eigenvectors of a Matrix.

4 Eigenvectors(Matrix([[4, -, 6], [2,, 6], [2, -, 8]])); 9 2 2, K (4.4) Test if the Matrix M is orthogonal. M:=Matrix([[sqrt(0)*/0, -sqrt(0)/0], [sqrt(0)/0, sqrt (0)*/0]]); M := K (4.5) Isrthogonal(M); true Solve the system defined by Matrix M and Vector v. (4.6) M:=Matrix([[,,, -], [,,, ], [, -2,, -], [4,, 8, -]]); M := K K2 K 4 8 K (4.7) v:=vector([0,,, 0]); v := 0 0 (4.8) LinearSolve(M, v); (4.9)

5 (4.9) K 5 2 K2 Construction of simple Matrices and Vectors, extraction of submatrices, transposition, basic arithmetic and computation of inner products can be done directly without use of commands in the LinearAlgebra package. u := Vector([,]); u := (4.0) v := Vector([5,7]); v := 5 7 (4.) u+v; 6 0 (4.2) u.v; A := Matrix([[,],[5,7]]); 26 (4.) A := 5 7 (4.4) B := Matrix([[,],[,]]); B := (4.5) A+2*B; (4.6) A.B;

6 (4.7) A.u; 0 26 (4.8) A^(-); K K 8 (4.9) A^%T; 5 7 (4.20) Details For more information including: a complete list of the routines in the LinearAlgebra package the supported data structures and data types the different sets of commands based on usage scenario: casual use or programming use the LinearAlgebra[Modular] subpackage for performing dense linear algebra computations in Z/m. the LinearAlgebra[Generic] subpackage for computing with generic implementations of algorithms for linear algebra over fields, Euclidean domains, integral domains and rings. see the Details of the Linear Algebra Package help page. See Also LAIndex examples/la_syntax_shortcuts - LinearAlgebra package shortcuts. VectorCalculus - a collection of commands that perform multivariate and vector calculus operations.